Systems and Methods for Battery Performance Monitoring and Management Using DiscreteTime StateSpace Overpotential Battery Models
Disclosed are systems, methods, and other implementations, including a method for monitoring and managing battery performance that includes deriving a representation of diffusion overpotential behavior for a lithiumion battery according to a discretetime statespace approximation of a convolutiondefined diffusion (CDD) model for the lithiumion battery, and determining behavior of the lithiumion battery according to the discretetime statespace approximation of the convolutiondefined diffusion (CDD) model for the lithiumion battery.
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This application claims the benefit of, and priority to, U.S. Provisional Application No. 63/412,956, entitled “SYSTEMS AND METHODS FOR BATTERY PERFORMANCE MANAGEMENT USING OVERPOTENTIAL BATTERY MODELS” and filed Oct. 4, 2022, the content of which is incorporated herein by reference in its entirety.
BACKGROUNDLithiumion battery (LIB) models form the basis of most battery management systems (BMS). The BMS depends on accurate models to perform state estimation. Battery models can also be used to provide insight into cell degradation, which could allow for advanced degradationreduction techniques. There are two (2) popular model types: physicsbased models (PBM) and equivalent circuit models (ECM). PBMs are highly accurate and offer insight into internal cell processes, but face challenges in realtime use. ECMs are especially popular in electric vehicle (EV) applications for their simplicity and speed. More recently, fractionalorder models (FOM) and physicallymeaningful ECMs have been proposed that can quantify electrochemical overpotentials in the cell. The frequencyvarying impedance is modelled with a FOM such as the Randles circuit, whose parameters can be used to provide insight into degradation modes in the cell. FOMs include a constantphase element (CPE). The CPE is defined by a fractionalorder transfer function, which is shown to accurately capture the charge transfer and diffusion overpotentials. FOMs face challenges in BMS implementation because frequencydomain data is difficult to obtain in realtime.
SUMMARYDescribed herein is a proposed battery performance management framework for lithiumion batteries (LIB) model that characterizes the overpotentials generated by ion transport and the change in the opencircuit voltage in a cell. The proposed framework models the ohmic, charge transfer, and diffusion overpotentials, and uses the standard resistorcapacitor pair elements for the ohmic and charge transfer dynamics. Diffusion dynamics are derived from the convolutiondefined diffusion (CDD) model. The proposed framework uses a discretetime statespace approximation of the CDD model (using a “recedinghorizon diffusion,” or RHD, approach) model. To identify the parameters, battery voltage, current and an optimization process are used. Experimentation results show that battery voltage can be tracked with 99% accuracy. Furthermore, the overpotential information allows cell internal parameters to be monitored. This helps determine the maximum power output capabilities of the cell, which varies with the degradation level. The RHD model of the proposed framework may be integrated into battery management systems in, for example, electric vehicles, electrical grids (e.g., that include PV arrays), as well as other applications, and can be used in standard state estimation techniques.
Thus, in some variations, a method for monitoring and managing battery performance is provided that includes deriving a representation of diffusion overpotential behavior for a lithiumion battery according to a discretetime statespace approximation of a convolutiondefined diffusion (CDD) model for the lithiumion battery, and determining behavior of the lithiumion battery according to the discretetime statespace approximation of the convolutiondefined diffusion (CDD) model for the lithiumion battery.
Embodiments of the method may include at least some of the features described in the present disclosure, including one or more of the following features.
Determining the behavior of the lithiumion battery may include applying input current to the lithiumion battery, capturing voltage response of the lithiumion battery resulting from applying the input current, and determining diffusion overpotential for the lithiumion battery based on analysis of the voltage response according to the discretetime statespace approximation of the convolutiondefined diffusion (CDD) model.
Deriving the representation of the diffusion overpotential behavior may include deriving the discretetime statespace approximation based on a recursive formulation using diffusion state data computed by the model extending back to a predetermined number of instances defining a finite time horizon.
Deriving the representation of the diffusion overpotential behavior may include deriving a diffusion related constant, A_{D}, at steady state for the lithiumion battery, with the lithiumion battery comprising a nickelmanganese cobalt (NMC) cell, according to:
where 1/β is the maximum stoichiometric added lithium, v_{M }is the molar volume of the NMC cell, S is the active surface area, F is Faraday's constant, and D is the Lithiumion diffusion coefficient.
Deriving the representation of the diffusion overpotential behavior may include determining the CDD model representing the diffusion overpotential behavior of the lithiumion battery as the product of a diffusion related constant A_{D}, and a convolution of a unit impulse response, g_{z}(t), with a timedependent diffusion state amplitude, ξ, for the lithiumion battery according to:
V_{D}(t)=A_{D}ξ(t)*g_{z}(t)
with the timedependent diffusion state amplitude, ξ, being determined based on a step change ΔI and a gradient of the opencircuit voltage (OCV) curve for the lithiumion battery, given by:
ξ_{n}(t)=ΔI(t_{n})∇V_{OC}√{square root over (t−t_{n})},
and with g_{z}(t) being defined as g_{z}(t)=√{square root over (t)}−√{square root over (t−Δt)}.
The method may further include deriving the discretetime statespace approximation according to a recursive relationship represented as:
x_{v}(t_{k+1})=A_{v}x_{v}(t_{k})+B_{v}u_{v}(t_{k})
V_{D}(t_{k})=C_{v}x_{v}(t_{k})
where x_{v}(t_{k}) is a diffusion state vector comprising diffusion state samples, u_{v}(t_{k}) is an input current vector applied to the lithiumion battery, C_{v }is a vector with values depending on a diffusion related constant A_{D}, A_{v }is a matrix with values depending on a relative sampling time for state samples of the discretetime statespace model, and B_{v }is a row vector with an entry that depends on an OCV gradient and differential current applied to the discretetime statespace model.
Deriving the discretetime statespace approximation according to the recursive relationship may include computing the diffusion overpotential V_{D }based on a current diffusion state sample and M preceding diffusion state samples, where M is a predetermined tunable value defining a computational horizon.
Computing the diffusion overpotential V_{D }may include weighing the diffusion states samples arranged in x_{v}(t_{k}) with diminishing weights, specified by the matrix A_{v}, to decrease the contribution of earlier computed diffusion states.
The nonzero diminishing weights specified in the matrix A_{v }may be computed according to
where m=1, 2, . . . , M, where M represents the number of samples in a computational horizon for computing the diffusion overpotential V_{D}.
The method may further include deriving based on the discretetime statespace approximation one or more battery performance metrics and/or battery degradation data.
The method may further include determining based on the battery degradation data one or more of, for example, state of health (SoH) of the lithiumion battery, and/or state of charge (SoC) for the lithiumion battery.
In some variations, a battery performance monitoring and management system is provided that includes one or more memory storage devices to store data and executable instructions, and a processorbased controller, coupled to the one or more memory storage devices. The processorbased controller is configured to derive a representation of diffusion overpotential behavior for a lithiumion battery according to a discretetime statespace approximation of a convolutiondefined diffusion (CDD) model for the lithiumion battery, and determine behavior of the lithiumion battery according to the discretetime statespace approximation of the convolutiondefined diffusion (CDD) model for the lithiumion battery.
In some variations, a nontransitory computer readable media is provided that includes computer instructions executable on a processorbased device to derive a representation of diffusion overpotential behavior for a lithiumion battery according to a discretetime statespace approximation of a convolutiondefined diffusion (CDD) model for the lithiumion battery, and determine behavior of the lithiumion battery according to the discretetime statespace approximation of the convolutiondefined diffusion (CDD) model for the lithiumion battery.
Embodiments of the system and the computer readable media may include at least some of the features described in the present disclosure, including at least some of the features described above in relation to the method.
Other features and advantages of the invention are apparent from the following description, and from the claims.
These and other aspects will now be described in detail with reference to the following drawings.
Like reference symbols in the various drawings indicate like elements.
DESCRIPTIONDescribed herein is a proposed battery performance management framework for lithiumion batteries (LIB) that uses a novel equivalent circuit models referred to as the ‘convolutiondefined diffusion’ (or CDD) and ‘recedinghorizon diffusion’ (or RHD). The CDD model accurately describes the diffusion overpotential using convolution, while the RHD model presents a recursive matrix formulation that approximates the CDD. It is believed that this is the first time that a discreterecursive statespace definition of diffusion is achieved with only one modelling parameter, and without fractionalorder computations. The RHD model is accurate, fast, and linked to the diffusion coefficient. A standard BMS can easily implement the RHD model to track electrochemical overpotentials in realtime. Verification of the proposed approach using simulated and experimental data shows that the RHD model is fast, accurate, and generalpurpose compared to standard RCpair circuits. It can be easily adapted to existing BMS state estimation techniques such as Kalman filters and can offer further insight into battery degradation. In some embodiments, more extreme temperatures and current rates can be integrated into the modeling framework. In various examples, subsampling techniques could be used to reduce the size of the diffusion state vector. Finally, the RHD model can be integrated with advanced diagnostics in a real EV or grid system. This can give a greater understanding of internal cell dynamics with small increases in computation time.
With reference to
The CDD model is represented by a circuit model 200 shown in
Details about the CDD and the RHD approximation will next be discussed. Under the CDD model, lithiumion diffusion is quantified with the diffusion constant A_{D}, given by:
where 1/β is the maximum stoichiometric added lithium, v_{M }is the molar volume of NMC, S is the active surface area, F is Faraday's constant, and D is the Lithiumion diffusion coefficient.
It has been shown that in a DNRC model (a battery equivalent circuit model that includes a diffusion component and N resistorcapacitor pairs) each current step change at time to t_{n}≤t incites a diffusion state ψ_{n}(t) defined as:
ψ_{n}(t)=ξ_{n}√{square root over (t−t_{n})} (2)
with a diffusion state amplitude ξ_{n }at step index n. The amplitude ξ_{n }depends on the value of the step change ΔI and the gradient of the opencircuit voltage (OCV) curve, and is given by:
ξ_{n}(t)=ΔI(t_{n})∇V_{OC}√{square root over (t−t_{n})} (3)
where,
where V_{OC }is the OCV, SoC is the state of charge, Δt is the sampling interval, and i is the cell current.
The DNRC diffusion overpotential is then given by:
where N_{step }is the total number of step changes. It is undesirable to use a sum of accumulated states. Since the DNRC model requires each current step change to be identified and labelled, for an arbitrarilyvarying cell current this formulation can be computationally prohibitive, hence the need for CDD.
To arrive at the CDD model, the input current is treated as a continuoustime system sampled with zeroorder hold (ZOH). The ZOHmodified cell current is given by the sum of rectangle pulses, denoted as the function rect(·),
for t>0. At each sampling index k, there are now two step changes in current corresponding to the rising and falling edge of the ZOH pulse. Thus, the diffusion overpotential becomes:
where ψ_{k}(t) and ψ′_{k}(t) are the rising and falling edge diffusion states due to the ZOH pulse at index k. Similarly, the diffusion state amplitude at the rising edge is given by,
ξ_{k}=i(t_{k})∇V_{OC}(t_{k})=ξ(t_{k}) (8)
which is equal and opposite at the falling edge, ξ_{k}=−ξ_{k}.
Note that ξ can now be described as a function of time. Substituting Equation (8) into Equation (2) yields the sum of the rising and falling edge states, as follows:
ψ_{k}(t)+ψ′_{k}(t)=ξ(t_{k})g_{z}(t−t_{k}) (9)
for t>t_{k}, where the unit impulse response is defined as:
g_{z}(t)=√{square root over (t)}−√{square root over (t−∇t)} (10)
The aggregated diffusion overpotential is therefore the infinite sum of the ZOHmodified responses, which represents convolution, denoted by *. Substituting Equation (9) into Equation (7) yields the CDD overpotential:
Equation (11) shows that the diffusion overpotential can be represented as the convolution of the diffusion state amplitude ξ with the unit impulse response g_{z}. Compared to the DNRC formulation, convolution is computed much faster given an arbitrary current input. Convolution is not suitable for realtime, however, because the entire current history must be stored. A recursive definition is thus preferred. The CDD model is linked to FOMs due to its fractionalorder impulse response. The Laplace transform G(s) of the continuoustime impulse response is given by:
which represents a semiintegral in the timedomain, expressible using fractional calculus.
The RHD model state equations are given by:
where the output is the sum of all overpotentials:
y(t_{k})=V_{s}(t_{k})+V_{ct}(t_{k})+V_{D}(t_{k}) (16)
The cell terminal voltage V_{o }is then given by:
V_{0}(t_{k})=V_{OC}(t_{k})−y(t_{k}) (17)
When N=2 RCpairs are used, the variables A_{l}, B_{l}, C_{l}, and D_{l }are given by standard NRC equations, namely:
It is to be noted that x_{l}∈R^{N }is the solution and charge transfer state vector, A_{l}∈R^{N×N }is a diagonal matrix, B_{l}∈R^{N }is a column vector, and C_{l}∈R^{N }and D_{l}∈R^{N }are row vectors. The variables x_{v}, A_{v}, B_{v}, and C_{v }describe the diffusion element in the RHD model and are formulated below. As noted earlier, the DNRC model is infeasible for long time scales due to the constantlygrowing state vector. Meanwhile, the CDD model requires the entire current history to be stored. These disadvantages are avoided by applying a recedinghorizon. All states beyond the horizon are assumed to saturate at some constant value. Only states within the horizon are tracked, and an offset term is used to store the saturated states.
The recursivediscrete form of a diffusion state is obtained from ZOH discretization of Equation (2). Due to the nonlinearity of squareroot dynamics, it can be shown that the discreterecursive definition is a piecewise function of the form:
For a continuouslyvarying current input, there is a current step at each k, so there is no longer a need to track the n^{th }current step. Thus, from Equations (3) and (14), the following relationship is defined:
ψ_{k}(t_{k+1})=b_{v}(t_{k})u(t_{k}) (20)
where b_{v}(t_{k})∈R^{2 }is a row vector dependent on the time step that introduces the OCV gradient and differential current into the RHD model:
b_{v}(t_{k})=∇V_{OC}(t_{k})√{square root over (Δt)}(1 1) (21)
For a horizon with a length of M time steps, the diffusion overpotential becomes:
where x_{v}∈R^{M+2 }is a column vector, A_{v}∈R^{(M+2)×(M+2) }is a square matrix, B_{v}(k)∈R^{(M+2)×2 }is a tall matrix with M+2 rows and 2 columns, Cv∈R^{M+2 }is a row vector, D_{v }is a scalar, and a_{m }is a scalar that is defined by the relative sampling time m, based on the relationship:
The parameter A_{D }in C_{v }is the only modelling parameter that must be identified. The step horizon M is treated as a tuning parameter that is fixed before implementation of the RHD model. Step horizon corresponds to a length of time, so for a 10 s horizon and a sampling interval of 0.1 s, M=100.
Dynamic processes occur in the state vector due to the offdiagonal of the matrix A_{v}. At each k, A_{v }advances the states to the next time step. One state saturates and another is initialized. Saturated states are stored in the offset term D_{v}. This offset term is an accumulated sum of the saturated values linked to OCV change. In contrast to the CDD model, the RHD model only requires a fixed number of states that are updated recursively in discretetime. Thus, a completely linear statespace definition is achieved.
To verify that battery degradation data derived using an RHDbased model provide a good approximation to the CDD model, a fivefold verification process was performed.
More particularly, in some embodiments, simulations of pulse perturbation are performed using varied diffusion coefficients in a coupled agglomeratescale and electrodescale continuum PBM for an NMC (nickelmanganese cobalt) cell. Diffusivities are in the range {0.2, 0.3, . . . , 1}×10^{−10 }while the nominal SoC ranges from 0.1 to 0.9. A CDD model module is then used to predict the diffusion coefficient. For simplicity, the PBM does not capture charge transfer dynamics, so no RC pairs are used in the CDD model (hence the name ‘CDD0RC’). The only modelling parameters are therefore R_{0 }and A_{D}. Apparent diffusivity {circumflex over (D)} is calculated from the CDD parameter A_{D }and the PBM parameters, according to:
where the parameters 1/β=0.55 (corresponding to the maximum stoichiometric added lithium, ε_{AM}=0.306 (corresponding to the volume fraction of active material), L_{agg}=1 μm (corresponding to the agglomerate size), S is the active surface area, F is Faraday's constant, and D is the Lithiumion diffusion coefficient.
In the second stage of the verification process, frequency analysis of the CDDsimulated data is performed at frequency analysis module 330. Frequency analysis of simulated CDD model data is performed by calculating the complex frequencyvarying impedance Z(s) according to
where s is the complex angular frequency, and V(s) and I(s) are the complex frequency spectra of the voltage and current. To obtain V(s) and I(s), the following example three operations may be performed:

 1. Simulate time domain pulse voltage responses, with CDD1RC model, lasting several minutes and with a sampling frequency 10 Hz. Pulses are used because they excite a wide range of frequencies.
 2. Apply Hamming window to reduce spectral noise.
 3. Perform discrete fast Fourier transform of the Hammingwindowed voltage and current data.
The Nyquist curve is a plot of the negative imaginary impedance −Im(Z) against real impedance Re(Z). A range of parameters are used, with R_{0}={0.02, 0.04, 0.08}Ω, R_{1}={0.01, 0.02, 0.03}Ω, C_{1}=1000 F, and A_{D}={9, 15, 25}×10^{−4 }A^{−1 }1 s^{−0.5}. To observe the effects of R_{0}, R_{1}, and A_{D}, one parameter is varied while the others are held constant. This allows the frequency behavior to be clearly observed. Performance of the second stage of the verification process demonstrates strong links to diffusion.
In the third stage, the CDD model yields strong parameter variation trends in experimental degradation data. In an example, experimental cell aging data is collected from three (3) commercial 2.7 Ah lithium NMC oxide cells (Panasonic NCR18650PF) held, for example, at 10° C. (e.g., in a temperature chamber 360). A battery cycler 350 implements some prespecified battery cycling protocol. For example, the battery cycler 350 is configured to cause cells to be degraded by lowvoltage cycling at 1 Crate, which can represent incomplete charging and high depthofdischarge. This usage profile could be common in portable electronics. There are 14 unique SoH in the range [0.78, 1]. At each SoH, unipolar charge pulses are applied to the cell at nine (9) SoC in the range [0.1, 0.9]. A degradation analysis module 370 records and processes the experimental data captured from the behavior of the batteries in response to stimuli applied to them (e.g., via the battery cycler 350).
The CDD1RC model parameters are then estimated from the voltage responses. Parameter estimation may be performed (at a parameter estimator module, which, in some embodiments, may be implemented at the degradation analysis module 370), for example, based on a scattersearch nonlinear global optimization process. This is represented as: minimize f(θ) subject to θ>0. Next, the following relationships are defined:
f(θ)=∥r∥_{2}^{2}+w∥r′∥_{2}^{2 }
r=y−ŷ(θ) (27)
where f is the objective function, y and ŷ are the observed and predicted data vectors, θ is the parameter vector, ∥r∥_{2}^{2 }is the sum of squared residuals, ∥r′∥_{2}^{2 }is the sum of squared residual differences, and the weighting coefficient w is set to w=1.
In the fourth stage, performed by a horizon analysis module 340, RHD simulation is made to approximate the CDD model for a sufficient horizon length. For example, in some embodiments, voltage response data is simulated using an RHD model 312 and the CDD model 310 using rectangle and sawtooth wave input waveforms (e.g., generated by a waveform input generator 302). Identical parameters are used in the 2RCpair models, but the RHD horizon length is varied from 10 s to 300 s, with example values of R_{0}=0.02Ω, R_{1}=0.1Ω, R_{2}=0.01Ω, C_{1}=7×10^{3 }F, C_{2}=1Ω10^{3 }F, and A_{D}=0.001 A^{−1}s^{−0.5}. The horizonlengthvarying error between the RHD approximation and CDD simulation is then examined for both waveforms.
Finally, in the fifth stage, the RHD model 312 is shown to increase prediction accuracy using experimental drivecycle data compared to NRC models (represented by block 314) and to track individual overpotentials.
To further verify the validity of the CDD model, the link between CDD and diffusion is demonstrated with PBMsimulated data. The aggregated diffusion coefficient of the cell can be determined from a physicsbased model (PBM). The CDDpredicted apparent diffusivity {circumflex over (D)} may be calculated from the diffusion constant A_{D }and the known PBM parameters. This is then compared with the true model diffusivity D.
As illustrated in
The second link to the diffusion overpotential is from frequency analysis of CDDsimulated data. In electrochemical impedance spectroscopy, the overpotentials are clearly observed in the frequency domain Nyquist curves. Only diffusion is known to affect the lowfrequency ‘tail’. Therefore, the frequency spectrum of the simulated CDD model should yield distinct behavior in the Nyquist curves.
Experimental aging data can be used to demonstrate how the CDD model can track cell parameters over the cell's lifetime. This is important for cell diagnostics. Different usage profiles may result in different parameter trends, which can inform optimal cell cycling conditions. Results for cells degraded at low temperature and SoC are shown in
The graphs 630 and 640 show CDD1RC modelling parameters plotted against SoH and SoC. There are very strong trends in the CDD1RC parameters, as shown in the graphs 630 and 640. Resistances increase as state of health (SoH) decreases, while capacitance decreases. The diffusion constant A_{D}, which varies inversely with diffusivity, is seen to increase as SoH decreases.
To further verify the RHD model, simulations of the RHD and CDD models are compared to demonstrate how the receding horizon approximates convolution. Two waveforms were used for comparison: sawtooth and current steps.
Experimental drive cycle data is used to compare realtime RHD model performance to that of conventional NRC models. The Crates for four distinct 2minute drive cycles (US06, urban dynamometer driving schedule (UDDS), LA92, and highway fuel economy test (HWFT)), applied at {0, 10, 25}° C. are shown in graph grouping 800 of
The second use of drive cycle data is for tracking overpotentials. Results of overpotential analysis using the LA92 drive cycles are shown in
Thus, as discussed above, and as demonstrated in the various simulations and experimentations, a novel ECM, referred to as the CDD model, was shown to capture the diffusion overpotential in LIB cells using convolution. For realtime implementation, the RHD model was formulated with a linear discretetime recursive statespace system. This is believed to the first linear representation of diffusion to use a single modelling parameter without fractionalorder computations. Verification using simulated and experimental data showed that the RHD model is fast, accurate, and generalpurpose compared to standard RCpair circuits. It can be easily adapted to existing BMS state estimation techniques such as Kalman filters to offer further insight into battery degradation. The proposed framework can be used in wider operating range than that demonstrated herein, and with higher optimization of the statespace system. More extreme temperatures and current rates can be used. Subsampling techniques could also be used to reduce the size of the diffusion state vector. Finally, the RHD model can be integrated with advanced diagnostics in a real electrical vehicle (EV), or in a grid system (as will be discussed below).
With reference next to
In some examples, determining the behavior of the lithiumion battery may include applying input current to the lithiumion battery, capturing voltage response of the lithiumion battery resulting from applying the input current, and determining diffusion overpotential for the lithiumion battery based on analysis of the voltage response according to the to the discretetime statespace approximation of the convolutiondefined diffusion (CDD) model. Deriving the representation of the diffusion overpotential behavior may include deriving the discretetime statespace approximation based on a recursive formulation using diffusion state data computed by the model extending back to a predetermined number of instances defining a finite time horizon. In various examples, deriving the representation of the diffusion overpotential behavior may include deriving a diffusion related constant, A_{D}, at steady state for the lithiumion battery, with the lithiumion battery comprising a nickelmanganese cobalt (NMC) cell, according to:
where 1/β is the maximum stoichiometric added lithium, v_{M }is the molar volume of the NMC cell, S is the active surface area, F is Faraday's constant, and D is the Lithiumion diffusion coefficient.
In some embodiments, deriving the representation of the diffusion overpotential behavior may include determining the CDD model representing the diffusion overpotential behavior of the lithiumion battery as the product of a diffusion related constant A_{D}, and a convolution of a unit impulse response, g_{z}(t), with a timedependent diffusion state amplitude, ξ for the lithiumion battery according to V_{D}(t)=A_{D}ξ(t)*g_{z}(t), with the timedependent diffusion state amplitude, ξ, being determined based on a step change ΔI and a gradient of the opencircuit voltage (OCV) curve for the lithiumion battery, given by ξ_{n}(t)=ΔI(t_{n})∇V_{OC}√{square root over (t−t_{n})}, and with g_{z}(t) being defined as g_{z}(t)=√{square root over (t)}−√{square root over (t−Δt)}. In such embodiments, the procedure may further include deriving the discretetime statespace approximation according to a recursive relationship represented as:
x_{v}(t_{k+1})=A_{v}x_{v}(t_{k})+B_{v}u_{v}(t_{k})
V_{D}(t_{k})=C_{v}x_{v}(t_{k})
where x_{v}(t_{k}) is a diffusion state vector comprising diffusion state samples, u_{v}(t_{k}) is an input current vector applied to the lithiumion battery, C_{v }is a vector with values depending on a diffusion related constant A_{D}, A_{v }is a matrix with values depending on a relative sampling time for state samples of the discretetime statespace model, and B_{v }is a row vector with an entry that depends on an OCV gradient and differential current applied to the discretetime statespace model.
Deriving the discretetime statespace approximation according to the recursive relationship may include computing the diffusion overpotential V_{D }based on a current diffusion state sample and M preceding diffusion state samples, where M is a predetermined tunable value defining a computational horizon. Computing the diffusion overpotential V_{D }may include weighing the diffusion states samples arranged in x_{v}(t_{k}) with diminishing weights, specified by the matrix A_{v}, to decrease the contribution of earlier computed diffusion states. The nonzero diminishing weights specified in the matrix A_{v }may be computed according to
where m=1, 2, . . . , M, where M represents the number of samples in a computational horizon for computing the diffusion overpotential V_{D}.
In some examples, the procedure may further include deriving based on the discretetime statespace approximation one or more battery performance metrics and/or battery degradation data. In such examples, the procedure may further include determining based on the battery degradation data one or more of, for example, state of health (SoH) of the lithiumion battery, and/or state of charge (SoC) for the lithiumion battery.
The use of the proposed RHD framework described herein to manage performance of cells (lithiumion batteries in the specific examples described herein, although the techniques can be similarly applied to other types of cells) can be used in relation to various applications that use rechargeable cells, including gridconnected solarphotovoltaic (PV) PV batteries, electrical vehicle batteries, etc. To illustrate how the RHD framework can be used in practical applications, consider the example of managing a PV battery.
Solar photovoltaics (PV) and lithiumion batteries (LIB) are dominant technologies with falling costs and more installations. Renewable sources like PV, however, suffer from decentralized generation and intermittency. Gridconnected PVbattery microgrids are illustrated in
To better understand the effects of different EMS strategies on system performance, three EMS frameworks are considered. Real data is used to simulate system performance using LL, PKS, and MPC EMS, representing increasing levels of sophistication. Costs from the grid, battery degradation, and PV credit are derived and assessed using different assumptions. The RHD overpotential ECM, discussed herein, is used to simulate the LIB cells. It can be shown that the benefits of the MPC EMS relative to LL or PKS depends on costing and modelling assumptions.
PV power generation is proportional to the solar irradiation P_{sun}[Wm^{−2}] received by a PV panel. The array is modelled from the cell level using:
where I is the cell current, V is the cell voltage, I_{p }is photocurrent (determined by P_{sun}), Is is diode saturation current, V_{T }is the diode thermal voltage, R_{sh }is shunt resistance, and n is the diode ideality factor. Series resistance is assumed negligible. A fourpanel array with area 4 m^{2}, N_{s}=60 cells per string, and N_{p}=8 parallel branches (2 branches per panel). Assume the cells are identical and operate in identical conditions, the array current and voltage are given by I_{array}=N_{p}I and V_{array}=N_{s}V. A DCDC converter may be used to control the cell voltage to obtain the maximumpowerpoint (MPP). Assume that the converter is lossless and that the PV system achieves MPP within one sampling interval.
The EMS uses the RHD model described herein (which links diffusivity with battery state of health) to model the battery. An overpotential ECM links diffusivity with battery state of health. The terminal voltage is given by:
v(t_{k})=V_{OC}(z,t_{k})−V_{s}(t_{k})−V_{ct}(t_{k})−V_{D}(t_{k}) (29)
where V_{OC }is the opencircuit voltage (OCV), z(t_{k}) is the cell state of charge (SoC), t_{k}is the time at step k, and V_{s}, V_{ct}, and V_{D }are the solution, charge transfer, and diffusion overpotentials. Standard NRC equations govern V_{s }and V_{ct }as follows:
with resistances R_{0 }and R_{1}, capacitance C_{1}, and sampling interval Δt. Note that each variable is a real integer when only 1 RCpair is used.
The diffusion overpotential is defined according to the RHD model, discussed above, using recursive matrix relationship, with M=2 (in this example):
x_{v}(t_{k+1})=A_{v}x_{v}(t_{k})+B_{v}u_{v}(t_{k})
V_{D}(t_{k})=C_{v}x_{v}(t_{k}) (31)
where the matrix A_{v }(whose dimensions is +2, which for M=2 is 4)
where x_{v}∈R^{M+2 }is a column vector, A_{v}(k) is a sparse square matrix with elements on the superdiagonal, B_{v }is a tall sparse matrix with M+2 rows and 2 columns, C_{v }is a row vector, and A_{D }is the diffusion constant.
Given a power demand P(t), the current and voltage must meet the demand at each time step, P(t_{k})=i(t_{k})v(t_{k}). Since the current input lags the state update vectors by one (1) step, Equations (29)(32) are substituted into the P(t) expression to obtain current from the quadratic equation:
where V_{st }is defined as V_{st}=V_{OC}(t_{k})−V_{ct}(t_{k})−V_{D}(t_{k}).
During discharge, V^{2}st>4R_{0}P(t_{k}) for the equation to yield a real value. This reflects the maximum power transfer limit. During charge, P<0 so the argument of the square root is always positive.
Battery degradation is modelled using simplified solidelectrolyte interface (SEI) layer dynamics at the negative electrode (NE), assumed to be the dominant degradation mechanism. This growth is quantified with a ‘sidereaction’ flux j_{s}(t_{k}) that does not contribute to intercalation mechanism. By assuming SEI formation dominance, sidereaction flux becomes directly proportional to the degradation rate. Total capacity Q_{T}(t_{k}) is then calculated using:
Q_{T}(t_{k})=Q_{T}(t_{k−1})+a_{n}A_{neg}_{neg}FΔtj_{s}(t_{k}) (34)
which shows degradation is exacerbated by high SoC and charging current.
The equations governing the evolution of parameters with state of health (SoH) are given by:
In the battery pack, it is assumed that all cells are identical, N_{s}=60 cells per series module, and N_{p}=14 branches in parallel so the pack voltage and current are given by v_{pack}=N_{s}v and i_{pack}=N_{p}i. Nominal pack voltage is V_{pack}^{nom}=50 V. Nominal cell voltage V_{cell}^{nom}=3.6 V is based on INR1865020R characteristics. Nominal pack capacity Q_{pack}^{nom}=10 kWh is comparable to average battery pack size in a residential household. Cell charging currents are limited to 3 A, discharge currents are limited by the load demand, and SoC is limited to the range [0.05, 0.95].
Using the above PV array modeling, which relied, in part, on the RHDbased modeling for the cells of the PV array (and without getting into the details of the specific performance of the various EMS control strategies), the PV array, battery overpotentials, and battery degradation were simulated. Costs were derived using marketbased analysis. The LL, PKS, and an original MPC EMS control strategies were formulated to understand the variation of net costs with a wide variety of simulation parameters. It was determined that modelbased EMS strategies were not necessarily superior to simple rulebased EMS if degradation costs were too high or the PV credit rates were too low. Still, the MPC EMS strategy delivered the most consistent results at the lowest costs.
Performing the various techniques and operations described herein may be facilitated by a controller device (e.g., a processorbased computing device). Such a controller device may include a processorbased device such as a computing device, and so forth, that typically includes a central processor unit or a processing core. The device may also include one or more dedicated learning machines (e.g., neural networks) that may be part of the CPU or processing core. In addition to the CPU, the system includes main memory, cache memory and bus interface circuits. The controller device may include a mass storage element, such as a hard drive (solid state hard drive, or other types of hard drive), or flash drive associated with the computer system. The controller device may further include a keyboard, or keypad, or some other user input interface, and a monitor, e.g., an LCD (liquid crystal display) monitor, that may be placed where a user can access them.
The controller device is configured to facilitate, for example, battery performance monitoring and management (e.g., based on discretetime statespace overpotential models). The storage device may thus include a computer program product that when executed on the controller device (which, as noted, may be a processorbased device) causes the processorbased device to perform operations to facilitate the implementation of procedures and operations described herein. The controller device may further include peripheral devices to enable input/output functionality. Such peripheral devices may include, for example, flash drive (e.g., a removable flash drive), or a network connection (e.g., implemented using a USB port and/or a wireless transceiver), for downloading related content to the connected system. Such peripheral devices may also be used for downloading software containing computer instructions to enable general operation of the respective system/device. Alternatively and/or additionally, in some embodiments, special purpose logic circuitry, e.g., an FPGA (field programmable gate array), an ASIC (applicationspecific integrated circuit), a DSP processor, a graphics processing unit (GPU), application processing unit (APU), etc., may be used in the implementations of the controller device. Other modules that may be included with the controller device may include a user interface to provide or receive input and output data. The controller device may include an operating system.
Computer programs (also known as programs, software, software applications or code) include machine instructions for a programmable processor, and may be implemented in a highlevel procedural and/or objectoriented programming language, and/or in assembly/machine language. As used herein, the term “machinereadable medium” refers to any nontransitory computer program product, apparatus and/or device (e.g., magnetic discs, optical disks, memory, Programmable Logic Devices (PLDs)) used to provide machine instructions and/or data to a programmable processor, including a nontransitory machinereadable medium that receives machine instructions as a machinereadable signal.
In some embodiments, any suitable computer readable media can be used for storing instructions for performing the processes/operations/procedures described herein. For example, in some embodiments computer readable media can be transitory or nontransitory. For example, nontransitory computer readable media can include media such as magnetic media (such as hard disks, floppy disks, etc.), optical media (such as compact discs, digital video discs, Bluray discs, etc.), semiconductor media (such as flash memory, electrically programmable read only memory (EPROM), electrically erasable programmable read only Memory (EEPROM), etc.), any suitable media that is not fleeting or not devoid of any semblance of permanence during transmission, and/or any suitable tangible media. As another example, transitory computer readable media can include signals on networks, in wires, conductors, optical fibers, circuits, any suitable media that is fleeting and devoid of any semblance of permanence during transmission, and/or any suitable intangible media.
Although particular embodiments have been disclosed herein in detail, this has been done by way of example for purposes of illustration only, and is not intended to be limiting with respect to the scope of the appended claims, which follow. Features of the disclosed embodiments can be combined, rearranged, etc., within the scope of the invention to produce more embodiments. Some other aspects, advantages, and modifications are considered to be within the scope of the claims provided below. The claims presented are representative of at least some of the embodiments and features disclosed herein. Other unclaimed embodiments and features are also contemplated.
Claims
1. A method for monitoring and managing battery performance comprising:
 deriving a representation of diffusion overpotential behavior for a lithiumion battery according to a discretetime statespace approximation of a convolutiondefined diffusion (CDD) model for the lithiumion battery; and
 determining behavior of the lithiumion battery according to the discretetime statespace approximation of the convolutiondefined diffusion (CDD) model for the lithiumion battery.
2. The method of claim 1, wherein determining the behavior of the lithiumion battery comprises:
 applying input current to the lithiumion battery;
 capturing voltage response of the lithiumion battery resulting from applying the input current; and
 determining diffusion overpotential for the lithiumion battery based on analysis of the voltage response according to the discretetime statespace approximation of the convolutiondefined diffusion (CDD) model.
3. The method of claim 1, wherein deriving the representation of the diffusion overpotential behavior comprises:
 deriving the discretetime statespace approximation based on a recursive formulation using diffusion state data computed by the model extending back to a predetermined number of instances defining a finite time horizon.
4. The method of claim 1, wherein deriving the representation of the diffusion overpotential behavior comprises: A D = 2 β v M S F D π where 1/β is the maximum stoichiometric added lithium, vM is the molar volume of the NMC cell, S is the active surface area, F is Faraday's constant, and D is the Lithiumion diffusion coefficient.
 deriving a diffusion related constant, AD, at steady state for the lithiumion battery, with the lithiumion battery comprising a nickelmanganese cobalt (NMC) cell, according to:
5. The method of claim 1, wherein deriving the representation of the diffusion overpotential behavior comprises:
 determining the CDD model representing the diffusion overpotential behavior of the lithiumion battery as the product of a diffusion related constant AD, and a convolution of a unit impulse response, gz(t), with a timedependent diffusion state amplitude, ξ, for the lithiumion battery according to: VD(t)=ADξ(t)*gz(t),
 wherein the timedependent diffusion state amplitude, ξ, is determined based on a step change ΔI and a gradient of the opencircuit voltage (OCV) curve for the lithiumion battery, given by: ξn(t)=ΔI(tn)∇VOC√{square root over (t−tn)},
 and wherein gz(t) is defined as gz(t)=√{square root over (t)}−√{square root over (t−Δt)}.
6. The method of claim 5, further comprising:
 deriving the discretetime statespace approximation according to a recursive relationship represented as: xv(tk+1)=Avxv(tk)+Bvuv(tk) VD(tk)=Cvxv(tk)
 where xv(tk) is a diffusion state vector comprising diffusion state samples, uv(tk) is an input current vector applied to the lithiumion battery, Cv is a vector with values depending on a diffusion related constant AD, Av is a matrix with values depending on a relative sampling time for state samples of the discretetime statespace model, and Bv is a row vector with an entry that depends on an OCV gradient and differential current applied to the discretetime statespace model.
7. The method of claim 6, wherein deriving the discretetime statespace approximation according to the recursive relationship comprises:
 computing the diffusion overpotential VD based on a current diffusion state sample and M preceding diffusion state samples, where M is a predetermined tunable value defining a computational horizon.
8. The method of claim 6, wherein computing the diffusion overpotential VD comprises:
 weighing the diffusion states samples arranged in xv(tk) with diminishing weights, specified by the matrix Av, to decrease the contribution of earlier computed diffusion states.
9. The method of claim 8, wherein the nonzero diminishing weights specified in the matrix Av are computed according to a m = 1 + 1 m, where m=1, 2,..., M, where M represents the number of samples in a computational horizon for computing the diffusion overpotential VD.
10. The method of claim 1, further comprising:
 deriving based on the discretetime statespace approximation one or more battery performance metrics and/or battery degradation data.
11. The method of claim 10, further comprising:
 determining based on the battery degradation data one or more of: state of health (SoH) of the lithiumion battery, or state of charge (SoC) for the lithiumion battery.
12. A battery performance monitoring and management system comprising:
 one or more memory storage devices to store data and executable instructions; and
 a processorbased controller, coupled to the one or more memory storage devices, configured to: derive a representation of diffusion overpotential behavior for a lithiumion battery according to a discretetime statespace approximation of a convolutiondefined diffusion (CDD) model for the lithiumion battery; and determine behavior of the lithiumion battery according to the discretetime statespace approximation of the convolutiondefined diffusion (CDD) model for the lithiumion battery.
13. The system of claim 12, wherein the processorbased controller configured to determine the behavior of the lithiumion battery is configured to:
 apply input current to the lithiumion battery;
 capture voltage response of the lithiumion battery resulting from applying the input current; and
 determine diffusion overpotential for the lithiumion battery based on analysis of the voltage response according to the discretetime statespace approximation of the convolutiondefined diffusion (CDD) model.
14. The system of claim 12, wherein the processorbased controller configured to derive the representation of the diffusion overpotential behavior is configured to:
 derive the discretetime statespace approximation based on a recursive formulation using diffusion state data computed by the model extending back to a predetermined number of instances defining a finite time horizon.
15. The system of claim 12, wherein the processorbased controller configured to derive the representation of the diffusion overpotential behavior is configured to: A D = 2 β v M S F D π where 1/β is the maximum stoichiometric added lithium, vM is the molar volume of the NMC cell, S is the active surface area, F is Faraday's constant, and D is the Lithiumion diffusion coefficient.
 derive a diffusion related constant, AD, at steady state for the lithiumion battery, with the lithiumion battery comprising a nickelmanganese cobalt (NMC) cell, according to:
16. The system of claim 12, wherein the processorbased controller configured to derive the representation of the diffusion overpotential behavior comprises:
 determine the CDD model representing the diffusion overpotential behavior of the lithiumion battery as the product of a diffusion related constant AD, and a convolution of a unit impulse response, gz(t), with a timedependent diffusion state amplitude, ξ, for the lithiumion battery according to: VD(t)=ADξ(t)*gz(t),
 wherein the timedependent diffusion state amplitude, ξ, is determined based on a step change ΔI and a gradient of the opencircuit voltage (OCV) curve for the lithiumion battery, given by: ξn(t)=ΔI(tn)∇VOC√{square root over (t−tn)},
 and wherein gz(t) is defined as gz(t)=√{square root over (t)}−√{square root over (t−Δt)}.
17. The system of claim 16, wherein the processorbased controller is further configured to:
 derive the discretetime statespace approximation according to a recursive relationship represented as: xv(tk+1)=Avxv(tk)+Bvuv(tk) VD(tk)=Cvxv(tk)
 where xv(tk) is a diffusion state vector comprising diffusion state samples, uv(tk) is an input current vector applied to the lithiumion battery, Cv is a vector with values depending on a diffusion related constant AD, Av is a matrix with values depending on a relative sampling time for state samples of the discretetime statespace model, and Bv is a row vector with an entry that depends on an OCV gradient and differential current applied to the discretetime statespace model.
18. The system of claim 17, wherein the processorbased controller configured to derive the discretetime statespace approximation according to the recursive relationship is configured to:
 compute the diffusion overpotential VD based on a current diffusion state sample and M preceding diffusion state samples, where M is a predetermined tunable value defining a computational horizon.
19. The system of claim 17, wherein the processorbased controller configured to compute the diffusion overpotential VD is configured to: a m = 1 + 1 m, where m=1, 2,..., M, where M represents the number of samples in a computational horizon for computing the diffusion overpotential VD.
 weigh the diffusion states samples arranged in xv(tk) with diminishing weights, specified by the matrix Av, to decrease the contribution of earlier computed diffusion states;
 wherein the nonzero diminishing weights specified in the matrix Av are computed according to
20. A nontransitory computer readable media comprising computer instructions executable on a processorbased device to:
 derive a representation of diffusion overpotential behavior for a lithiumion battery according to a discretetime statespace approximation of a convolutiondefined diffusion (CDD) model for the lithiumion battery; and
 determine behavior of the lithiumion battery according to the discretetime statespace approximation of the convolutiondefined diffusion (CDD) model for the lithiumion battery.
Type: Application
Filed: Oct 4, 2023
Publication Date: Apr 18, 2024
Applicant: The Trustees of Columbia University in the City of New York (New York, NY)
Inventors: Alan Gen Li (New York, NY), Matthias Preindl (New York, NY)
Application Number: 18/376,581